Control member with a balance wheel and a planar spiral for a watch or clock movement

ABSTRACT

A regulating device including a balance ( 1 ) and a plane hairspring ( 2 ) for a time piece movement, the plane hairspring ( 2 ) including a stiffened portion ( 8 ) in its outer turn ( 7 ) that is arranged to cause the deformations of the turns to be substantially concentric. The spacing (d) between a terminal portion of the outer turn ( 7 ) and the last-but-one turn ( 9 ) of the hairspring ( 2 ) is large enough for said last-but-one turn ( 9 ) to remain free radially during expansions of the hairspring ( 2 ) up to amplitudes corresponding substantially to the maximum angle of rotation of the balance ( 1 ) in the movement.

BACKGROUND

A. Field

The present invention relates to a regulating device having a balanceand a plane hairspring for a time piece movement.

B. Related Art

It is known that the turns of a plane hairspring deform eccentricallywhen the hairspring is in operation. This eccentric deformation of theturns, which is explained by the fact that the center of gravity of thehairspring does not correspond to the center of rotation of the balanceand hairspring assembly, disturbs the setting of the balance andhairspring assembly and makes it anisochronous.

The center of gravity of the hairspring could be returned arbitrarily tothe center of rotation of the balance by being shifted, however thatwould not solve the problem since during the working of the hairspringthe center of gravity would move and would therefore no longer coincidewith the initial center of gravity.

Two different solutions have been proposed for causing the center ofgravity and the center of rotation to coincide while a plane hairspringis working, thereby making the deformations of the turns concentric:

the Breguet hairspring with a so-called Philips curve in which an outercurve is moved into a second plane above the hairspring plane; and

the hairspring with angle strip as described in 1958 by Messrs. Emileand Gaston Michel in an article entitled “Spiraux plats concentriquessans courbes” [Concentric flat hairsprings without curves], published bySociété Suisse de Chronométrie.

The first solution amounts to modifying the initial plane hairspring sothat it becomes a hairspring occupying a plurality of planes. Thatsolution does not come within the ambit of the present invention whichrelates only to plane hairsprings.

The second solution consists in stiffening a determined portion of aturn by giving it the shape of an angle strip. The angle strip issituated either in the outer turn or in a central turn. Nevertheless,according to the authors of that solution, although angle strip in thecentral turn does indeed provide a significant improvement in terms ofthe isochronism of the balance and hairspring assembly, angle strip inthe outer turn does not give satisfaction. Said authors even abandonedthe hairspring with angle strip in the outer turn, in the belief thatthey had been wasting their time on this topic.

BRIEF SUMMARY OF THE INVENTION

The present invention seeks to improve the isochronism of a balance andhairspring assembly by stiffening a portion of the outer turn of thehairspring, and for this purpose the invention provides a regulatingdevice as defined in the appended claims as well as a timepiece, such asa watch, incorporating the above-mentioned regulating device.

The present invention is based on the observation that, contrary to theconclusions reached by the authors of the above-mentioned article“Spiraux plats concentriques sans courbes”, it is possible to improvesignificantly the isochronism of a balance and hairspring assembly bystiffening a determined portion of the outer turn of the hairspring,provided that the spacing between the terminal portion of the outer turnand the last-but-one turn of the hairspring is sufficiently large toensure that said last-but-one turn remains radially free duringexpansions of the hairspring up to amplitudes correspondingsubstantially to the maximum angle of rotation of the balance in themovement.

According to the present inventors, the reason why the solution usingangle strip in the outer turn as described in the above-mentionedarticle provided no improvement in terms of isochronism, stems from thefact that during large-amplitude expansions of the hairspring thelast-but-one turn came into abutment against the outer turn or against astud or an index pin associated with said outer turn, therebysignificantly disturbing the operation of the hairspring. The presentinventors have observed that by modifying the hairspring described inthe above-mentioned article in such a manner that expansion of thelast-but-one turn is not impeded by the last turn (the outer turn), norby its accessories such as the stud and the index pins, operation of thebalance and hairspring assembly becomes substantially isochronous.

The present invention also provides a method of designing a regulatingdevice comprising a balance and a plane hairspring, the method being asdefined in accompanying claim 13, with particular implementationsthereof being defined in the corresponding dependent claims.

DESCRIPTION OF THE DRAWINGS

Other characteristics and advantages of the present invention willappear on reading the following detailed description made with referenceto the accompanying drawings, in which:

FIG. 1 is a plan view of a regulating device according to a firstembodiment of the invention;

FIG. 2 is a plan view showing, by way of comparison, the hairspring of aconventional regulating device, in its rest position;

FIGS. 3 and 4 are plan views showing the hairspring of FIG. 2respectively in expansion and in compression, in a theoretical situationwhere the center of the hairspring is free, the outer end of thehairspring being taken as a fixed reference point;

FIG. 5 is a plan view showing the hairspring of the regulating device ofthe first embodiment of the invention in its rest position;

FIGS. 6 and 7 are plan views showing the hairspring of FIG. 5respectively in expansion and in compression, in a theoretical situationwhere the center of the hairspring is free, the outer end of thehairspring being taken as a fixed reference point;

FIG. 8 is a plan view showing the hairspring of a regulating deviceaccording to a second embodiment of the invention, together withaccessory elements thereof;

FIG. 9 is a diagrammatic plan view showing how a portion to be stiffenedof the outer turn of the hairspring of the regulating device of theinvention is determined;

FIGS. 10 to 12 are plan views showing variants of an intermediatehairspring shape obtained during a method of designing the regulatingdevice of the invention;

FIG. 13 is a diagrammatic plan view showing a theoretical expansion ofan intermediate hairspring obtained in the design method of theinvention and in which the terminal portion of the outer turn remains tobe corrected; and

FIG. 14 is a diagrammatic plan view showing how the terminal portion ofthe outer turn of the hairspring shown in FIG. 13 is corrected so as toenable the last-but-one turn to remain radially free during,expansionsof the hairspring up to amplitudes that correspond substantially to themaximum angle of rotation of the associated balance.

DETAILED DESCRIPTION OF THE PREFERRED

With reference to FIG. 1, a regulating device for a time piece movementaccording to the invention comprises a balance 1 and a flat hairspring 2in the form of an Archimedes' spiral. The inner end 3 of the hairspring2 is fixed to a collet 4 driven onto the shaft of the balance 1 and istherefore continuously subjected to the rotary torque from the balance1. In known manner, the rotary shaft of the balance and hairspringassembly turns in bearings (not shown). The outer end 5 of thehairspring 2 is fixed to a stationary part of the movement, typicallythe balance-cock, via a fixing member 6 called “stud”.

According to the invention, the hairspring 2 has in its outer turn 7 astiffened portion 8 that is arranged to cause the deformations of theturns to be substantially concentric during expansions and compressionsof the hairspring 2. This stiffened portion 8 is constituted by aportion of the strip forming the hairspring having a greater thickness ein the plane of the hairspring than does the remainder of the strip.This thickness e is large enough relative to the thickness e₀ of theremainder of the strip to confer stiffness to the stiffened portion 8that is much greater than that of the remainder of the strip. Thus,during expansions and compressions of the hairspring, the stiffenedportion 8 hardly deforms at all, and therefore does not participate inthe deformation of the turns. In the example shown, the thickness e ofthe stiffened portion 8 varies, with its minimum, at the ends of thestiffened portion, being equal to the thickness e₀ of the remainder ofthe strip and its maximum, in the center of the stiffened portion, beingequal to three times the thickness e₀ of the remainder of the strip.Nevertheless, as will be apparent in the following, the thickness e ofthe stiffened portion could equally well be constant or could vary onlyin the terminal portions of the stiffened portion. The extra thicknesspresented by the stiffened portion 8 relative to the remainder of thestrip is preferably situated exclusively on the outer side of the lastturn 7 so as to ensure that it cannot come into contact with thelast-but-one turn, identified by reference 9. The way in which thestiffened portion 8 is arranged along the hairspring 2 is explainedbelow with reference to the method of the invention.

As explained in the introduction to this application, the turns in aconventional hairspring deform eccentrically since the center of gravityof the hairspring does not correspond with its geometrical center. Thegeometrical center of the hairspring is the center of the frame ofreference in which its spiral is defined. It is situated on the axis ofrotation of the balance and hairspring assembly. FIG. 2 shows, by way ofillustration, a conventional hairspring in the form of an Archimedes'spiral in its rest position, together with the associated frame ofreference (O, x, y) and the center of gravity G₀ of the hairspring.FIGS. 3 and 4 show the same hairspring respectively after it has beenexpanded by one revolution (360°) and after it has been compressed byone revolution by applying a pure torque to the inner end of thehairspring, the outer end of the hairspring being taken as a fixedreference point. The term “pure torque” is used to mean that the innerend of the hairspring is free, i.e. the theoretical circumstance isassumed whereby the axis of the balance and hairspring assembly is freeto move parallel to the plane of the hairspring, or in other words isnot held by bearings. As can be seen, during such expansion andcompression, the geometrical center O′ of the hairspring as representedby a point inside a circle moves mainly along the axis (O, x), towardsnegative values for x during expansion and towards positive values for xduring compression, and therefore no longer coincides with the center Oof the frame of reference. In practice, since the geometrical center ofthe hairspring cannot move because of the constraint imparted by thebearings on the shaft of the balance and hairspring assembly, the way inwhich the turns deform during expansion or compression of the hairspringis necessarily eccentric, and not concentric as shown in FIGS. 3 and 4.

In the present invention, the function of the stiffened portion 8 is tobring the center of deformation of the hairspring 2 to the geometricalcenter of the said hairspring. The center of deformation of thehairspring is the center of gravity of the elastic portion of thehairspring, i.e. of the portion of the hairspring other than itsstiffened portion 8. FIGS. 5, 6, and 7 show the hairspring 2 of theregulating device of the invention respectively at rest, expanded afterapplying a pure torque of the same amplitude as in FIG. 3 (the outer endof the hairspring being taken as a fixed reference point, as in FIG. 3),and in compression after applying a pure torque having the sameamplitude as in FIG. 4 (the outer end of the hairspring being taken as afixed reference point, as in FIG. 4). It can be seen that thegeometrical center O′ of the hairspring 2 hardly moves and remains incoincidence with the center O of the frame of reference during suchexpansion and compression. This implies that in practice the constraintexerted by the bearings on the shaft of the balance and hairspringassembly is sufficiently small for the deformations of the turns toremain substantially concentric, as in the theoretical circumstances ofFIGS. 6 and 7. This leads to a significant improvement in theisochronism of the balance and hairspring assembly, which will workpurely in torque in its bearings without being subjected to disturbingforces due to reaction from the bearing supports.

With reference again to FIG. 1, according to another characteristic ofthe invention, the spacing or radial distance d between a terminalportion of the outer turn 7 and the last-but-one turn 9 is large enoughto ensure that this last-but-one turn 9 remains radially free duringexpansions of the hairspring 2 up to amplitudes correspondingsubstantially to the maximum angle of rotation of the balance 1 in themovement. The term “maximum angle of rotation” is used herein to meanthe maximum angle that the balance wheel 1 is liable to reach duringnormal conditions of operation of the movement. This angle is determinedin particular by the force from the mainspring (barrel spring) of themovement. It is less than the knocking angle. In a typical embodiment ofthe invention, the maximum angle of rotation is slightly less than theknocking angle and is equal to about 330°. It is recalled that theknocking angle is defined as being the angle of rotation of the balancefrom which knocking occurs, i.e., typically, the angle from which theimpulse-pin of the balance comes into contact with the outer face of ahorn of the fork of the escapement pallets.

In other words, the radial spacing or distance d is large enough toensure that during normal operation of the movement, the last-but-oneturn 9 cannot come into contact either with the outer turn 7 or with thestud 6. Since the expansions (and naturally also the compressions) ofthe last-but-one turn 9 are not impeded at any time during normaloperation of the movement, the deformations of the turns always remainconcentric, thereby leading to a significant improvement in theisochronism of the balance and hairspring assembly.

In practice, in order to retain a safety margin, the terminal portion ofthe outer turn 7 can be positioned far enough away from the last-but-oneturn 9 so as to ensure that the latter cannot reach said terminalportion even during expansions of the hairspring going as far asamplitudes corresponding to the absolute maximum angle of rotation ofthe balance, i.e. the knocking angle.

FIG. 8 shows a second embodiment of the invention, in which theregulating device comprises in particular a hairspring 2′ having astiffened outer turn portion 8′, a stud 6′ for fastening the outer end5′ of the hairspring, and an index, of which only the pins 10 are shown,for adjusting the active length of the hairspring 2′. The stiffenedouter turn portion 8′ presents a double bend 11 in its central portion.This double bend 11 enables the terminal portion of the outer turn 7′,from the double bend 11 to the outer end 5′, firstly to be far enoughaway from the last-but-one turn 9′ to ensure that neither this terminalportion nor its accessories such as the stud 6 and the pins 10 canimpede the expansions of the last-but-one turn 9′, and secondly to havea generally circularly-arcuate shape of center O that is adapted torotation of the index. Nevertheless, in a variant, the index and itspins 10 could be omitted.

There follows a description of the method for designing the hairsprings2 and 2′.

Firstly, an Archimedes' spiral is defined in a frame of reference (O, x,y), by the known formula:r(θ)=r ₀ +pθwhere r₀ and p are predetermined constants and where r and θ are polarcoordinates in the frame of reference (O, x, y).

This spiral is given a strip thickness e₀ in the plane of the spiral anda strip height h₀ perpendicular to the plane of the spiral. These valuese₀ and h₀ are constant over the entire length of the spiral.

The coordinates (x_(G), y_(G)) of the center of gravity G of thehairspring obtained in this way are calculated as follows:

$x_{G} = {\frac{1}{L}{\int_{0}^{L}{x{\mathbb{d}s}}}}$$y_{G} = {\frac{1}{L}{\int_{0}^{L}{y{\mathbb{d}s}}}}$where L is the length of the hairspring and ds is the incremental lengthalong the hairspring.

Using these equations:

x = r cos  θ y = r sin  θ,  and${ds} = {\sqrt{{r^{2}\left( {d\;\theta} \right)}^{2} + ({dr})^{2}} = \sqrt{{r^{2}\left( {d\;\theta} \right)}^{2} + {p^{2}\left( {d\;\theta} \right)}^{2}}}$the coordinates x_(G) and y_(G) can also be written as follows:

$x_{G} = {\frac{1}{L}{\int_{0}^{2\pi\; N}{r\;\cos\;\theta\sqrt{{r^{2}\left( {d\;\theta} \right)}^{2} + {p^{2}\left( {d\;\theta} \right)}^{2}}}}}$$y_{G} = {\frac{1}{L}{\int_{0}^{2\pi\; N}{r\;\sin\;\theta\sqrt{{r^{2}\left( {d\;\theta} \right)}^{2} + {p^{2}\left( {d\;\theta} \right)}^{2}}}}}$where N is the real number of turns of the hairspring.

The unbalance of the hairspring is then deduced as calculated at thecenter of gravity G:{right arrow over (b)}_(G)=m{right arrow over (OG)}where m is the total mass of the hairspring: m=

e₀h₀L where

is the mass density of the hairspring, and the vector {right arrow over(OG)} defined by the points O and G (which are assumed to be situated inthe same plane parallel to the plane of the hairspring) has as itsmodulus:

$a = \sqrt{x_{G}^{2} + y_{G}^{2}}$

A portion of the outer turn that is to be made inactive will then bedetermined so that the unbalance {right arrow over (b)}_(G) which isresponsible for the anisochronism of the balance and hairspring assemblybecomes zero. This portion of the outer turn will then be reinforced sothat it loses its elasticity and constitutes a “dead zone” that does notparticipate in the deformations of the outer turn.

To eliminate the unbalance {right arrow over (b)}_(G), the portion ofthe turn that is to be made inactive must itself present an unbalance{right arrow over (b)} equal to the unbalance {right arrow over(b)}_(G). This turn portion is necessarily such that the point G liesbetween the point O and said turn portion and has an angular extentβ₂−β₁=2α (cf. FIG. 9) that is symmetrical about the axis passing throughthe points O and G.

By assuming that this outer turn portion is a circular arc of meanradius (half-thickness radius) R_(e), of center O, and of mass Δm, themodulus of its unbalance {right arrow over (b)} is equal to R_(e)Δm,where:Δm=

e ₀ h ₀ ΔL with ΔL=R _(e)(β₂−β₁)=2R _(e)αwhich gives:R_(e) Δm=ma=

e ₀ h ₀ Lai.e.:2R_(e) ²α=Lawhence:

$\alpha = \frac{La}{2R_{e}^{2}}$and:β₁=β_(G)−αβ₂=β_(G)+αwhere β_(G) is the angular position of the point G:β_(G)=Arctan (y _(G) /x _(G)).

The section of the outer turn portion delimited by the angles β₁ and β₂is then reinforced by giving this outer turn portion a thickness e inthe plane of the hairspring that is greater than the thickness e₀, e.g.that is equal to three times the thickness e₀. FIG. 10 shows thehairspring as obtained in this way with its stiffened portion beingidentified by reference 8″.

Preferably, in order to avoid or at least reduce any risk of thehairspring breaking during fabrication or while in operation at theradially extending straight ends 12 of the stiffened portion 8″, theshape of the stiffened portion 8″ is corrected so as to soften thetransition between the latter and the remainder of the strip. Thiscorrection of the shape of the stiffened portion 8″ is typicallyperformed as follows:

Initially, a function f=e(θ) is selected that is representative of thethickness in the plane of the hairspring of the corrected stiffenedportion as a function of polar angle θ. This function f is convex andcontinuous, and presents a minimum equal to the thickness e₀ at each ofthe two ends of the stiffened portion.

Thereafter, the angular extent δ₂−δ₁ of the corrected stiffened portionis calculated. This angular extent δ₂−δ₁ includes the angular extentβ₂−β₁ of the stiffened portion 8″ shown in FIG. 10; in other words δ₁<β₁and δ₂>β₂ (cf. FIGS. 9 and 10).

In order to determine the angles δ₁ and δ₂, it is assumed that thecorrected stiffened portion is to deform in the same manner as the turnportion defined by said angles δ₁ and δ₂ in the hairspring of FIG. 10.Assuming that the stiffness of the stiffened portion 8″ is infinite,which is the ideal theoretical value, the deformation of the turnportion of the hairspring of FIG. 10 between the angles δ₁ and δ₂ isequal to the sum of the respective deformations of the turn portionsbetween the angles δ₁ and β₁ and between the angles β₂ and δ₂. Thecomponent along the axis (O, x) of this deformation can thus be writtenas follows:

$D_{x}^{e_{0}} = {\frac{12M}{h_{0}e_{0}^{3}}\left\lbrack {{\int_{\delta_{1}}^{\beta_{1}}{y{\mathbb{d}s}}} + {\int_{\beta_{2}}^{\delta_{2}}{y{\mathbb{d}s}}}} \right\rbrack}$where M is the moment of deformation or torque applied to the hairspringand, as mentioned above, y=r sin θ with r=r₀+pθ. As for the component ofthe deformation of the corrected stiffened portion along the axis (O,x), this can be written as follows:

$D_{x}^{f} = {\frac{12M}{h_{0}}{\int_{\delta_{1}}^{\delta_{2}}\frac{y{\mathbb{d}s}}{f^{2}}}}$The components of the above-mentioned deformations along the axis (O, y)can be ignored since they are negligible and of the same order ofmagnitude as production errors. In order to reduce the number ofvariables, the angle δ₂−δ₁ is caused to be symmetrical about the axispassing through the points O and G. This makes it possible to define asingle variable φ equal to β_(G)−δ₁ and to δ₂−β_(G). This variable φ iscalculated by equating the deformation components D_(x) ^(e) ⁰ and D_(x)^(f):

${\frac{1}{e_{0}^{3}}\left\lbrack {{\int_{\beta_{G^{- \varphi}}}^{\beta_{1}}{y{\mathbb{d}s}}} + {\int_{\beta_{2}}^{\varphi + \beta_{G}}{y{\mathbb{d}s}}}} \right\rbrack} = {\int_{\beta_{G} - \varphi}^{\varphi + \beta_{G}}\frac{y{\mathbb{d}s}}{f^{3}}}$

To solve the above equation, it is possible to perform an iterativecalculation starting from a given value for φ, that is large enoughcompared with the length of the stiffened portion 8″, and thendecreasing this value φ step by step until the two deformationcomponents D_(x) ^(e) ⁰ and D_(x) ^(f) become close enough to eachother. Typically, the iteration algorithm is stopped as soon as:|D _(x) ^(e) ⁰ −D _(x) ^(f)|<εwhere:

$ɛ = \frac{10^{- 5}\left( {{D_{x}^{e_{0}}} + {D_{x}^{f}}} \right)}{2}$Once the final value for φ has been determined, the stiffened portion isredrawn by giving it the variable thickness e(θ)=f between the angles δ₁and δ₂.

By way of example, a function f that is particularly suitable for thevariable thickness of the corrected stiffened portion is givenherebelow:

$f = {e_{0} + {e_{0}\left\{ {1 + {\cos\left\lbrack {2\pi\frac{\left( {\theta - \beta_{G}} \right)}{\delta_{2} - \delta_{1}}} \right\rbrack}} \right\}}}$This function f presents a minimum equal to the thickness e₀ at bothends of the corrected stiffened portion, and a maximum equal to threetimes the thickness e₀ in the center of the corrected stiffened portion.This function f has the advantage of being convex over the entire lengthof the corrected stiffened portion, i.e. at all points along saidlength, thereby minimizing any risk of breakage. FIG. 11 shows thehairspring obtained after the step of correcting the stiffened portionwith such a function.

Nevertheless, the person skilled in the art will observe that otherconvex functions f can also be suitable. By way of example, FIG. 12shows a hairspring obtained after the step of correcting the stiffenedportion using a function f such that the thickness e of the correctedstiffened portion, identified by reference 8′″, is constant over theentire length thereof with the exception of terminal portions 13 whereit decreases continuously towards the ends 14 of said portion 8′″.

It should be observed that when corrected in this way by means of eitherone of the above-mentioned functions, the stiffened portion presents theadvantage not only of reducing the risk of the hairspring breaking, butalso of presenting stiffness that is greater than that of the stiffenedportion 8″ shown in FIG. 10 because its angular extent can be calculatedon the basis of infinite stiffness for the stiffened portion 8″.

Once the stiffened portion has been corrected, maximum expansion of thehairspring is simulated, e.g. by means of a finite element calculation,said maximum expansion corresponding to the maximum angle of rotation ofthe balance, and the shape of the terminal portion of the outer turn iscorrected so that the terminal portion is far enough away from thelast-but-one turn to ensure, as explained above, that neither theterminal portion nor its accessory elements (stud, index pins) canimpede expansion of the last-but-one turn. This correction of the shapeof the terminal portion is nevertheless sufficiently small to avoidsignificantly modifying the unbalance of the hairspring and of thestiffened portion. By way of illustration, FIG. 13 shows the theoreticalexpansion of a hairspring having a stiffened portion in its outer turn,but in which the terminal portion of the outer turn, whose shape has notyet been corrected, is not far enough away from the last-but-one turn.As can be seen, the last-but-one turn, identified by reference 16,extends beyond the position of the end 17 (considered as being fixed) ofthe outer turn, which means that in practice the last-but-one turn 16will come into abutment against said end 17 or against the stud to whichsaid end 17 is connected.

To move the terminal portion of the outer turn away from thelast-but-one turn, the following steps can be performed (cf. FIG. 14):

-   -   A first point P₁ is defined on the radial axis passing through        the outer end of the hairspring, which point is situated beyond        the last-but-one turn when the hairspring is at maximum        expansion, i.e. when the balance has reached its maximum angle        of rotation (to do this, a theoretical configuration is assumed        in which the last-but-one turn is not impeded radially and is        therefore maximally expanded, as in the example of FIG. 13), and        located at a distance from said last-but-one turn that is equal        to about one pitch of the spiral, for example (likewise when the        hairspring is at maximum expansion). In FIG. 14, the position of        the outer end of the hairspring is identified by reference P₀        and the position of the point of intersection between the        last-but-one turn and the above-mentioned radial axis when the        hairspring is at maximum expansion is identified by reference P′        (said position P′ is also shown in FIG. 13).    -   A second point P₂ is defined that is situated on the outer turn        at the end of the stiffened portion that is farther from the        outer end of the hairspring.

The first and second points P₁ and P₂ are interconnected by a circulararc 18 that is tangential to the outer turn at the second point P₂. Thecenter of this circular arc 18 is identified in FIG. 14 by the referenceO″.

-   -   A third point P₃ is defined on the circular arc 18 between the        first and second points P₁ and P₂, the third point P₃ being such        that the length of the segment of the circular arc 18 delimited        by the second and third points P₂ and P₃ is equal to the length        of the initial turn segment 19 delimited by the second point P₂        and the initial outer end P₀ of the hairspring.    -   Two angles T₁ and T₂ are defined in a frame of reference of        center O″ and whose axes are parallel to the axes in the frame        of reference (O, x, y). The angle T₂ is the angle made by the        straight line segment [O″, P₂] with the abscissa axis of the        frame of reference of center O″. The angle T₁ is such that the        length of the portion of the circular arc 18 delimited by the        angles T₁ and T₂ is equal to the length of the portion of the        initial turn segment 19 that is delimited by the angles δ₁ and        δ₂ in the frame of reference (O, x, y).    -   The circular arc 18 between the second and third points P₂ and        P₃ is given a thickness identical to that of the initial turn        segment 19. This thickness therefore varies between the angles        T₁ and T₂ and is constant elsewhere. The function fc=e(θ″)        defining said varying thickness between the angles T₁ and T₂ as        a function of the polar angle θ″ in the above-mentioned frame of        reference of center O″ is obtained by replacing the parameters        β_(G), δ₁ and δ₂ respectively by the parameters θ₀″, T₁ and T₂        in the function f defining the varying thickness of the portion        of the initial turn segment 19 that is delimited by the angles        δ₁ and δ₂, where θ₀″=(T₁+T₂)/2. Thus, for example, for a        function:

$f = {e_{0} + {e_{0}\left\{ {1 + {\cos\left\lbrack {2\pi\frac{\left( {\theta - \beta_{G}} \right)}{\delta_{2} - \delta_{1}}} \right\rbrack}} \right\}}}$

-   -    the function fc is given by:

${fc} = {e_{0} + {e_{0}\left\{ {1 + {\cos\left\lbrack {2\pi\frac{\left( {\theta^{''} - \theta_{0}^{''}} \right)}{T_{2} - T_{1}}} \right\rbrack}} \right\}}}$

The turn segment delimited by the second and third points P₂ and P₃ thenconstitutes the corrected terminal portion of the outer turn.

In a variant, in order to obtain the hairspring shown in FIG. 8, thefollowing steps can be performed for moving the terminal portion of theouter turn away from the last-but-one turn:

-   -   A point is defined on the outer turn in the stiffened portion,        typically at the center thereof.    -   The terminal portion of the hairspring extending from said point        is offset radially outwards, by giving the inner side of said        terminal portion the shape of a circular arc of center O and the        outer side of said terminal portion a shape that gives said        terminal portion the same thickness as that of the corresponding        initial terminal portion. This thickness thus varies between        said point and the angle δ₁ and is constant between the angle δ₁        and the outer end of the hairspring. The radial spacing between        this terminal portion and the last-but-one turn is selected to        be large enough to ensure that the last-but-one turn cannot        reach said terminal portion even when the hairspring is at        maximum expansion.    -   The above-mentioned terminal portion is connected to the        remainder of the stiffened portion by a straight line portion so        as to form the double bend 11. This straight line portion is of        sufficient thickness so as to avoid diminishing the stiffness of        the stiffened portion, for example its thickness is equal to        three times the thickness e₀ of the hairspring outside the        stiffened portion.

The hairspring of the regulating device of the invention is typicallymade of silicon. It can be fabricated in various ways, for example usingthe method described in document EP 0 732 635.

The present invention is described above purely by way of example. Itwill be clearly apparent to the person skilled in the art thatmodifications can be made without going beyond the ambit of theinvention. In particular, although it is preferable for the stiffenedportion to be formed by increasing the thickness of the strip formingthe hairspring in the plane of the hairspring, it is possible in avariant to increase the height of the strip (i.e. the thickness of thestrip perpendicularly to the plane of the hairspring). Naturally, undersuch circumstances, the height of the strip needs to be increased by arelatively large amount in order to obtain stiffness comparable to thatobtained in the case of a varying thickness in the plane of thehairspring. In another variant, both the thickness of the strip in theplane of the hairspring and the height of said strip could be varied.

1. A plane hairspring for a regulating device of a timepiece movementand having a plurality of turns including an outer turn, the planehairspring including along the outer turn a stiffened portion arrangedto cause deformations of the turns of the hairspring to be substantiallyconcentric when the hairspring is in operation in the timepiecemovement, wherein said stiffened portion is a portion of strip ofthickness in the plane of the hairspring greater than a thickness of aremainder of the strip forming the hairspring, and the extra thicknessdefined by the stiffened portion relative to the remainder of the stripis situated exclusively along on an outer side of the outer turn.
 2. Theplane hairspring according to claim 1, wherein the thickness in theplane of the hairspring of the stiffened portion varies over the entirelength of the stiffened portion as a convex and continuous function andpresents a minimum substantially equal to the thickness of the remainderof the strip at the two ends of the stiffened portion and a maximum thatis greater than the thickness of the remainder of the strip between saidtwo ends.
 3. The plane hairspring according to claim 1, wherein thethickness in the plane of the hairspring of the stiffened portion issubstantially constant over the entire length of said stiffened portion.4. The plane hairspring according to claim 1, wherein the thickness inthe plane of the hairspring of the stiffened portion is substantiallyconstant over the entire length of said stiffened portion except interminal portions where, respectively, the thickness decreasescontinuously towards the ends of said stiffened portion.
 5. The planehairspring according to claim 1, wherein the height of the hairspring issubstantially constant over the entire length of said hairspring.
 6. Aplane hairspring for a regulating device of a timepiece movement andhaving a plurality of turns including an outer turn, the planehairspring including along the outer turn a stiffened portion arrangedto cause deformations of the turns of the hairspring to be substantiallyconcentric when the hairspring is in operation in the timepiecemovement, wherein said stiffened portion is a portion of strip ofthickness in the plane of the hairspring greater than a thickness of aremainder of the strip forming the hairspring, and the thickness of thestiffened portion in the plane of the hairspring varies over the entirelength of the stiffened portion as a convex and continuous function andpresents a minimum substantially equal to the thickness of the remainderof the strip at two ends of the stiffened portion and a maximum that isgreater than the thickness of the remainder of the strip between saidtwo ends.
 7. A timepiece movement including a regulating deviceincluding a balance and the plane hairspring according to claim
 1. 8.The timepiece movement according to claim 7, wherein the spacing betweena terminal portion of the outer turn and the last-but-one turn of thehairspring is large enough for said last-but-one turn to remain freeradially during expansions of the hairspring up to amplitudescorresponding substantially to a maximum angle of rotation of thebalance in said movement.
 9. The timepiece movement according to claim8, wherein the maximum angle of rotation of the balance in said movementis equal to 330°.
 10. The timepiece movement according to claim 8,wherein the spacing between the terminal portion of the outer turn andthe last-but-one turn of the hairspring is large enough for saidlast-but-one turn to remain free radially during expansions of thehairspring up to amplitudes corresponding substantially to the knockingangle of the balance in said movement.
 11. A method of designing a planehairspring for a regulating device of a timepiece movement including:defining a plane hairspring of constant strip thickness; calculating theunbalance of said plane hairspring; calculating a portion of the outerturn of said plane hairspring having the same unbalance as the planehairspring; and stiffening said outer turn portion.
 12. The methodaccording to claim 11, wherein said stiffening step includes stiffeningsaid outer turn portion sufficiently so that said outer turn portionsubstantially does not deform during operation of the hairspring. 13.The method according to claim 11, wherein said stiffening step includesincreasing the thickness of said outer turn portion in the plane of thehairspring.
 14. A method of designing a plane hairspring for aregulating device of a timepiece movement including: defining a planehairspring of constant strip section; calculating the unbalance of saidplane hairspring; calculating a portion of the outer turn of said planehairspring having the same unbalance as the plane hairspring; andvarying the thickness, in the plane of the hairspring, of the stripforming the hairspring between an angle δ₁ and an angle δ₂ such thatδ₁<β₁ and δ₂>β₂, where β₂−β₁ is the angular extent of said portion ofthe outer turn, the thickness being caused to vary in accordance with apredetermined function f presenting a minimum substantially equal to thethickness of the remainder of the strip at the angles δ₁ and δ₂, thefunction f and the angles δ₁ and δ₂ being selected so that thedeformation of the turn portion delimited by the angles δ₁ and δ₂ issubstantially the same as the deformation which would occur if thethickness of the strip between the angles δ₁ and β₁ and between theangles β₂ and δ₂ were the same as that of the remainder of thehairspring and if, between the angles β₁ and β₂, the stiffness of theouter turn were equal to a predetermined value, greater than that of theremainder of the strip.
 15. The method according to claim 14, whereinsaid predetermined value is infinite.
 16. The method according to claim14, wherein the predetermined function f is convex and continuous.
 17. Amethod of making a plane hairspring for a regulating device of atimepiece movement, including designing the hairspring according to themethod as defined in claim 14 and then fabricating said hairspring. 18.The method according to claim 11, further including providing a spacingbetween a terminal portion of the outer turn and the last-but-one turnof the hairspring, said spacing being large enough for said last-but-oneturn to remain free radially during expansions of the hairspring up toamplitudes corresponding substantially to the maximum angle of rotationof a balance in said movement.
 19. The method according to claim 18,wherein said step of providing a spacing includes: defining a firstpoint on the radial axis passing through the outer end of saidhairspring having said stiffened portion, the first point being situatedbeyond the last-but-one turn of said hairspring when said last-but-oneturn is expanded by an amplitude corresponding to the maximum angle ofrotation of the balance; defining a second point on the outer turn;interconnecting the first and second points by a circular arc that istangential to the outer turn at the second point; defining a third pointon the circular arc between the first and second points, the third pointbeing such that the length of the segment of the circular arc delimitedby the second and third points is equal to the length of the initialturn segment delimited by the second point and the initial outer end ofthe hairspring; and giving a thickness in the plane of the hairspring tothe circular arc between the second and third points that is identicalto the thickness of the initial turn segment, the resulting turn segmentbetween the second and third points constituting a corrected terminalportion of the outer turn.
 20. The method according to claim 19, whereinthe second point is situated at the end of the stiffened portion that isfurther from the outer end of the hairspring.
 21. The method accordingto claim 18, wherein said step of providing a spacing includes: defininga point on the outer turn in the stiffened portion; offsetting theterminal portion of the hairspring extending from said point radiallyoutwards by giving the inner side of said terminal portion acircularly-arcuate shape the center of which is the geometrical centerof the hairspring and the outer side of said terminal portion a shapethat gives said terminal portion a thickness in the plane of thehairspring that is identical to the thickness of the correspondinginitial terminal portion of the outer turn; and connecting the terminalportion with the remainder of the stiffened portion by a connectionportion that forms a double bend.
 22. A method of making a planehairspring for a regulating device of a timepiece movement, includingdesigning the hairspring according to the method as defined in claim 11and then fabricating said hairspring.